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От чего на самом деле зависит треиие? Естественные определяющие параметры в механике контактного взаимодействия и физике трения
В.Л. Попов
Берлинский технический университет, Берлин, 10623, Германия
Известно, что коэффициент трения в общем случае зависит от большого числа системных параметров и параметров, определяющих внешнюю нагрузку. Шарль Кулон экспериментально показал, что статический коэффициент трения может зависеть от времени, нормальной силы, размера контакта, природы контактирующих материалов и от наличия промежуточных смазочных слоев. В случае коэффициента трения скольжения он описал зависимости от скорости скольжения и размера контакта. Последующие исследования показали, что коэффициент трения очень чувствителен к наличию колебаний (в том числе и автоколебаний). Несмотря на практическую важность проблемы трения, до сих пор не сформулированы обобщенные законы или эмпирические процедуры для измерения и представления зависимости силы трения как минимум от следующих четырех параметров: скорости, нормальной силы, формы (а тем самым и размера) контактирующих тел и времени. В настоящей статье обсуждается вопрос, каким образом можно уменьшать размерность пространства определяющих параметров и определять ограниченное множество «естественных определяющих параметров» процесса трения. Показано, что одним из таких естественных определяющих параметров является глубина индентирования (или относительное сближение) контактирующих тел. Также обсуждаются другие «кандидаты» на роль естественных определяющих параметров.
Ключевые слова: коэффициент трения, обобщенный закон трения, закон Амонтона-Кулона, механика контактного взаимодействия, влияние вибраций на трение
What does friction really depend on? Robust governing parameters in contact mechanics and friction
V.L. Popov
Berlin University of Technology, Berlin, 10623, Germany
It is known that the coefficient of friction generally depends on a large number of system and loading parameters. Already Coulomb presented experimental evidence that the static coefficient of friction may depend on time, on normal force, on the contact size, on the nature of contacting materials, and on the presence of intermediate lubricant layers. For the sliding coefficient of friction, he observed the dependence on the sliding velocity as well as the force and size dependencies. Later research has shown that the friction coefficient is very sensitive to the presence of oscillations (including self-excited vibrations). In spite of the practical importance of the problem, no generalized laws of friction or empirical procedures for measuring and representing the law of friction have been developed so far, which included at least the following four parameters: contacting body velocity, normal force, shape (and thus implicitly size), and time. In the present paper, we discuss the question of how the dimension of space of governing parameters can be reduced and if a small set of "robust governing parameters" of friction can be identified. We argue that one of such robust governing parameters is the indentation depth (or relative approach) of contacting bodies and discuss further candidates for the role of robust governing parameters.
Keywords: coefficient of friction, generalized law of friction, Amontons-Coulomb law, contact mechanics, influence of vibrations on friction
1. Introduction
Static and sliding friction are phenomena whose understanding is indispensable for the construction of safe and energy-saving designs. Knowledge of exact law of friction is of interest for countless applications, for example clutches, brakes, tires, bush and ball bearings, combustion engines, hinges, gaskets, castings, machining, cold forming, ultra-
sonic welding, electrical contacts, and many others [1, 2]. Friction is a phenomenon that people have been interested in for over hundreds and even thousands of years and still today remains in the mainstream of the development of new products and technologies.
However, in contrast to many other areas of engineering design, in most engineering applications the simplest Amon-
© Popov V.L., 2015
tons' law of friction is used stating that the force of friction is directly proportional to the normal force [3]. Every tribo-logist knows that this is only a very rough "zero approximation". Already Coulomb knew that the coefficient of static friction may depend on time, normal force, and on the contact size of the system while the sliding coefficient of friction generally depends on velocity, normal force and size as well as on many other factors [4, 5]. Even if many of these dependencies are relatively weak, they may have a major influence on the behaviour of tribological systems. Thus, even a weak velocity dependence can lead to development of dynamic instabilities and stick-slip movement [2].
In the past there have been many attempts to understand why the law of friction by Amontons and Coulomb is approximately valid [6-8], and such attempts continue to this day [9]. However, in the last time, the research becomes more focused on the deviations of the law of friction from the simple Amontons law [10-12]. The deviations from the simple Coulomb's law of friction can be related either to the dependence of the coefficient of friction on various system parameters or to more far reaching changes in the friction law. For example, since Coulomb, tribologists differentiate between the static and sliding friction. However, this differentiation is only relative. In reality, some preliminary sliding always occurs already under forces smaller than the critical force of gross slip. The phenomenon of preliminary sliding is of the fundamental importance for any applications where movements with small amplitudes are in play. These include the nanopositioning systems as well as systems under vibrations with small amplitudes. Corresponding empirical laws have been developed first for nano-positioning applications [13-15]. It was recently argued that the preliminary sliding may be just a manifestation of partial sliding under tangential loading [16, 17]. Dieterich, Rice and Ruina [18-20] went further and replaced the notions of "static" and "kinetic" friction with the unifying concept of rate- and state-dependent friction.
While there exist many investigations of friction in various particular conditions, the attempts to work out unifying concepts, which allow to construct "generalized laws of friction" which could be used by engineers in practical applications, are still rare. A successful example for formulation of such generalized laws provides the friction of elastomers. Based on the hypothesis of Grosch [21] about the rheologi-cal nature of elastomer friction, a very powerful "master curve procedure" has been developed [2, 22], which allows a unifying representation of the coefficient of friction as function of sliding velocity and temperature in the form of one single "master curve" in appropriate coordinates. The master curve procedure is based on the hypothesis that the coefficient of friction does depend only on the product of the velocity and relaxation time of the elastomer, while the temperature goes into the dependence only over the relaxation time. Such parameter combinations, which the coeffi-
cient of friction really depend on, will be called "robust governing parameters".
However, velocity and temperature are not the only parameters which friction of elastomers—to continue with this example—does depend on. As already shown by Coulomb and illustrated by simple one-dimensional simulations in [23, 24], the coefficient of friction also depends on the normal force and on the size of the contact configuration. In paper [24], a generalized master curve procedure was suggested and verified experimentally, which includes a unifying description of dependence of the coefficient of friction on velocity, temperature, and normal force. In [24], it was argued that one of the "robust governing parameters" could be the indentation depth. In the present paper, we discuss in a more detail this parameter as well as other candidates for the role of robust governing parameters in friction.
2. Indentation depth as a robust governing parameter of contact configuration
If a rigid body of an arbitrary shape is pressed against a homogeneous elastic half-space then the resulting contact configuration is only a function of the indentation depth d. For example, in the case of the Hertz-type contact of a rigid parabolic indenter with the curvature radius R pressed against an elastic half-space, the contact configuration is uniquely determined by the contact radius
a =4Rd. (1)
The contact radius does depend on the shape of the indenter and the indentation depth but it does not depend on elastic properties of the medium. This is a very general property which is valid for any shape of indenter in contact with an elastomer with arbitrary linear rheology. Let us illustrate this statement with a simple example of indentation of a parabolic indenter into an elastic body and into a linear viscous fluid. The equilibrium condition for an elastic continuum reads
Au + —^ Vdivu = 0, (2)
1 - 2v
where v is Poisson's ratio. The corresponding "equilibrium" equation for a linearly viscous fluid reads as follows:
Aii + —^ Vdivu = 0. (3)
1 - 2v
For "zero" initial conditions (plane surface of the medium before the indentation), the integration in time of (3) leads to (2). Thus, the equations determining the displacement field will be identical both for the elastic body and the fluid, and it is easy to see that the boundary conditions (given normal displacements inside the contact area and zero normal stress outside the contact area) will be also identical. It follows that both the contact radius and the complete form of the free surface will be identical in these cases too. This general behavior was recognized by Lee [25] and Radok [26] who used this property to formulate a generalized pro-
cedure of functional equations. The procedure is general and is applicable to arbitrary profiles. In [27] it was applied and verified numerically for fractal rough surfaces.
While the dependence of the contact configuration only on the indentation depth is an exact statement for a normal indentation of a rigid indenter into a linearly elastic continuum, it remains approximately valid also in other situations. For example, in the case of a non-linear elastic material with the power law stress-strain relationship e = e0 (/// )n, it was found [28] that the penetration depth is given by
, \(n—1) ,=^l2n+1 )
2n
(4)
In the case of the linearly elastic (n = 1) medium we reproduce the Hertz result (1) while in the general case there will be some correction factor of the order of unity. In the limiting case n = «>, the dependence (4) reads a ~ 1.65 JRd.
For tangential contact with a viscoelastic medium it is not exactly correct but remains qualitatively correct. The standard rough estimation of the contact configuration in this case [29] is as follows. Due to tangential motion, the effective elastic modulus will correspond to the frequency v/a, where v is the sliding velocity and a is the contact radius. However, as the relation between the indentation depth and the contact radius does not depend on the elastic modulus, this relation will remain (at least in the frame of this rough estimation) unchanged independently of the sliding velocity.
There are further indications that the contact configuration is a function of solely the indentation depth—with the same "qualitative precision"—even for spatially heterogeneous systems. For example, in [30], it was shown that in the case of an elastic medium with a thin coating, the equation (1) remains valid provided the properties of the coating differ not too strong from the properties of the underlying material. The same was shown in [31] for the case of multilayer materials. In [32], it was argued that this is equally valid for media which are heterogeneous in the lateral direction (along the contact plane).
We thus can summarize that the relation between the "contact configuration" and the indentation depth is relatively insensitive to material properties or the state of motion. Up to a factor of the order of unity, it remains the same, as in the case of the normal contact with a homogeneous elastic half-space.
Along with the contact configuration, all contact properties including the real contact area, the contact size, the contact stiffness, as well as the rms value of the surface gradient in the contact area will be unambiguous functions of the indentation depth. The indentation depth is thus a convenient and robust "governing parameter" for contact and frictional properties of media of various physical nature.
Let us now discuss another aspect of the law of friction— the preliminary sliding. In a series of recent papers devoted to the problem of presliding in stick-slip drives, it was shown
that the phenomenon of preliminary sliding can be easily and precisely understood in terms of partial sliding of curved elastic contacts. These results have been summarized in [33]. It was shown that the value of the presliding distance can be predicted precisely in dependence of materials properties of both contacting bodies, the size of the driving sphere, and the normal force just by assuming that the presliding distance is nothing else than the displacement till the start of the gross slip. This assumption was further exploited in simulations and experiments on the influence of in-plane and out-of-plane oscillations on friction. In [34], a theoretical analysis of the influence of the in-plane oscillations has been carried out. In [35, 36], these theoretical predictions have been verified in a wide range of sizes of the contacting bodies and for different normal forces. Thus, the hypothesis of the nature of the presliding as a pure partial sliding in the contact area can be considered as confirmed for both microscopic and macroscopic systems. The direct implication of this fact is that the presliding, too, is governed by the indentation depth. Indeed, the theoretical prediction [37] for the maximum tangential displacement till the start of the gross sliding is given by
(5)
where
E =
f 1 — v2
1 — v
2 \
—1
G =
2 — v, 2 — v2
—1
4G
4G2
E1 and E2 are the Young moduli of contacting bodies, Vj
and v2 are their Poisson's ratios,
and G1 and
G2 are the
shear elastic moduli. For isotropic media, the ratio E*/G* is on the order of unity. For example, if one of the bodies is rigid and the other is incompressible, this ratio is equal to E* / G* = 1.5. From this it follows that for a frictional contact with the coefficient of friction y, the maximum tangential displacement to the onset of complete sliding is determined solely by the indentation depth and has the order of magnitude of yd. This result is independent of the form of the bodies in contact and is valid for arbitrary bodies of revolution and even for randomly rough fractal surfaces. The latter fact was validated by direct numerical simulation of tangential contact of rough surfaces [38, 39]. Again, the indentation depth turns out to be the determining (governing) parameter of the frictional law.
Finally, consider the influence of normal oscillations on friction. It is obvious that in this case the indentation depth is also the main governing parameter of this process. In particular, the static coefficient of friction becomes zero if the amplitude of normal oscillations becomes equal to the indentation depth (so that the sample starts to "jump").
With the above argumentation we tried to substantiate the following thesis: Independently of the type of contact (elastic, viscoelastic, plastic), the velocity, and the particular considered property (coefficient of friction, presliding, in-
fluence of vibrations), the indentation depth is a parameter which provides the most robust information about the "contact configuration" in a broad sense, and therefore determines the law of friction. Thus, this quantity is surely one of the "robust governing parameters" of friction which we search for.
However, the contact configuration is unambiguously determined by the indentation depth only under assumption of the given topography of the contacting bodies. The properties which determine the surface topography are discussed in the next section.
3. Surface gradient and the size of microcontacts as main representative surface parameters
In [24], it was argued that at least for linearly elastic elastomers in most practical situations there are the largest and the smallest scales in the power spectrum of the surface roughness which determine their contact and frictional properties. The long wavelength part of the spectrum determines the stiffness of the contact while the short wavelength part determines the surface gradient playing the most important role in friction. Indeed, both in a plastic contact and in a viscoelastic contact in the region of the plateau of the coefficient of friction (as function of the sliding velocity), almost all "micro-asperities" have the "one-sided contact". Under this condition, the coefficient of friction is just equal to some weighted average of the surface gradient (slope) Vz, measured in the whole contact area. Of course, the average slope may depend on the normal force. Even if this dependence is very weak, we will discuss it later in this paper.
As has been shown already by Archard in 1950th [7] the main effect of changing the normal force is the changing number of asperities coming into contact, while the local conditions in the real contact area, including the surface gradient, depend only weakly on the normal force. Thus, the pure "rheological" contribution to the coefficient of friction—independently on whether we have to do with linearly or nonlinearly elastic or plastic body—is given by the characteristic surface gradient in the area of contact. Another important contribution to the force of friction can be the "adhesive" contribution which was introduced already by Coulomb [5] and later was considered by Bowden and Tabor as the main contribution in the case of metallic partners [6]. Recently, it was suggested that shearing of surface layers may give an additional contribution to the force of friction of elastomers, too [40, 41]. It is easy to see that this contribution is also governed by the surface gradient. Indeed, one of the most robust results of the contact mechanics of rough surfaces states that the real contact area A is determined by the following equation: F
A - 2—j—. (6)
EVz
This result was first found by Hyun et al. [42] and was confirmed in numerous subsequent studies. If, for the sake of
simplicity, we consider a square rigid block of the size L in a contact with an elastic half-space and assume that the normal force is large enough so that the contact stiffness has already achieved its plateau [43], then the normal force is given by the equation F — E Ld and Eq. (6) takes the form
A - 2Vd- (7)
This is again a "pure geometrical" relation which does not contain elastic properties of the system and thus is independent of the details of the process under consideration. It shows that the determining parameters of the real contact area are the size of the system, the indentation depth, and the surface gradient.
The main physical question which remains is: what is the surface gradient? For fractal surfaces this is a poorly defined property which depends on the accuracy with which the surface topography is measured. On can easily see this from the following simple estimation. The average gradient of the surface profile (Vz2) can be expressed as an integral of the spectral power density of the surface profile C(q) according to
(Vz 2 )-J C(q)q3dq.
(8)
It is known that many natural surfaces have the property of fractality, and for randomly self-affine, fractal surfaces the spectral power density is known to be a power function of the wave vector q, that is
C2D(q) =const • (q/qo)
-2 H-2
(9)
where H is the Hurst exponent and q0 is some reference wave vector [44]. For typical Hurst exponents smaller than one, the resulting integral (8) diverges at the upper limit of integration. This means that for a true fractal surface (without an upper cut-off wave vector), the surface gradient is infinitely large. In practice, of course, there is always some upper cut-off wave vector qmax and the surface gradient is determined by one or two orders of magnitude of wave vectors at and below qmax. In other words, for typical fractal surfaces, the friction force is determined by the roughness components with the largest wave vectors (or the smallest scale of the system). One can say that understanding friction is equivalent to understanding the nature of this smallest relevant scale. This statement is not limited to the spectral representation of fractal surfaces. It is sufficient that the characteristic surface gradient is increasing toward the smaller space scales. The above said means that the question about the coefficient of friction is essentially replaced by the question of determining the "relevant maximum wave vector", which is not just a geometrical but in most cases a physical problem.
Further essential parameter is the "typical size" a of the "microcontacts". This size determines the characteristic frequency of interaction between the bodies and thus the effective complex modulus of their interaction. In the approximation of Greenwood and Williamson [45], this size is given
Fig. 1. Contact between an elastomer and a rigid conical indenter which is moved tangentially with the velocity v (a); rheological model for a viscoelastic medium (b)
by a - ^/Rh, where R is the radius of curvature of asperities and h is the rms roughness. Using the simple estimation Vz - (h/R)1/2 (see [2]), it can also be written in the following form:
h (10)
a ■
Vz
The characteristic frequency w can then be estimated as
v vVz
to — —. (11) a h
Thus, we anticipate that the coefficient of friction should be a function of dimensionless combinations of the following parameters: indentation depth d and sliding velocity v as loading parameters, apparent contact size L as system parameter, rms roughness h and the relevant rms surface gradient Vz as roughness parameters, relaxation time t as material parameter.
4. Examples of generalized laws of friction
To get feeling how the parameters discussed above determine the coefficient of friction, it is instructive to analyze several particular examples of generalized laws of friction.
As an example, we consider a rigid wedge-shaped indenter of the form
z = g (x) = c|x| (12)
which is pressed into a viscoelastic foundation to a certain depth d and is moved tangentially with the velocity v (Fig. 1, a).
In this case, the coefficient of friction is evaluated as follows [29]:
F = c 2(cvT/d)-1/2(cvt/d)2
Fn ~ 1 + 1/2(cvr/ d )2 ' By taking into account that c = |vz|, and (cvt)/d - vt/a - WT, this result can be rewritten in the form
(13)
u = vzt
vt
(14)
In a similar contact of a sphere of radius R1 with the Kelvin body we have [29]
£[2 - 3£- 2£3 + 2(1 + £2)3/2]
u = vz-
with
[i -e3+(i+?2)3/2]4/3
(15)
vt
vt
(2R1d )
1/2
Thus, the coefficient of friction again has the general form (14). In a contact of a cone with a Maxwell body, the coefficient of friction is given by [29]
^ dl (cvt) - 2(1 - e~dl (c VT ) ) + ln(2 - e~dl (c v ) ) ' dl (cvt) - ln(2 - e~d/( cVT)) !
(16)
which is obviously again of the form (14).
Till now, we considered simple "one-asperity" shapes of indenters. However, as the dependence (14) contains only local geometrical parameters of asperities, which according to the general ideas of Archard [7], do depend on the loading conditions only weakly, this will be valid also for multicontact configuration, at least qualitatively. We thus come to the conclusion that form (14) may be very general.
Let us discuss the dimensionless parameter entering Eq. (14). There are two parameters Vz and a characterizing roughness and one further parameter, relaxation time t characterizing the rheology of the material. We note that two parameters are the minimum with which a rough surface can be characterized. For example, the Greenwood and Williamsion model uses the following two parameters: the rms roughness and the radius of curvature of asperities. However, in the most cases, the relevant curvatures and roughnesses "on the relevant scale" cannon be measured straightforwardly, just because we often do not know exactly the physical processes determining the smallest "relevant scale". They therefore will be in most cases used as empirical fitting parameters. However, in this case it is more sensible to use as empirical fitting parameters the parameters Vz and a which, according to Eq. (14), straightforwardly determine the coefficient of friction.
Of course, the assumed independence of the parameters Vz and a on the loading parameters is valid only approximately. Both these parameters depend on the actual "contact configuration". The latter, however, as argued in Sect. 2, depends solely on the indentation depth d. We thus can combine Eq. (14) with the argument of Sect. 2 in the following form:
vt(T )
u = vz (d )¥
a (d )
(17)
where we denoted explicitly Vz(d) and a(d) as (relatively weak) function of only indentation depth, while the relaxation time t(T) is a material parameter which does not depend on the contact configuration but may depend on the
temperature T. This form of the law of friction allows to suggest a new generalized master curve procedure. Equation (17) can be identically rewritten in the form
ln y = ln Vz(d) + ln fx
x [exp(ln v + ln t(T) - ln a(d))], (18)
or introducing a new function O(-) = ln f (exp(-)), in the form
ln y-ln Vz(d) = ®(ln v + ln t(T)-ln a(d)). (19) This equation states that if we plot the logarithm of the coefficient of friction as function of the logarithm of velocity, for different values of the indentation depth d and the temperature T, all curves will "look the same" and it is possible to move them by some vertical shift factor ln Vz (d) depending solely on the indentation depth as well as horizontally by the shift factor ln t(T) - ln a(d) which is the additive superposition of a function of d and a function of T.
Of course, in practice, the controlling (governing) parameter of the external loading is in most cases the normal force and not the indentation depth. The indentation depth generally will depend on the normal force, the size or the shape of the body, elastic properties, and for viscoelastic bodies also on the sliding velocity. However, all these influence factors are lying in a realm of the macroscopic contact mechanics and mechanical design and are principally solvable with standard techniques of contact mechanics. The whole "tribology", on the contrary, is completely contained in the phenomenological equation (19).
The suggested form (18) of the generalized law of friction is, of course, subjected to some restrictions. For example, we have assumed that the temperature is an independent external parameter. This assumption is only valid as long as the local "flash temperatures" are of no importance. However, it was shown in [46] that the general form (19) remains valid even under consideration of flash temperatures if a temperature dependent shift in the vertical direction is allowed.
Finally let us mention that the coefficient of friction is a kinetic property which under nonstationary conditions can be described with differential equations containing additional state variables [18-20]. However, as it was argued in [39] and [23, 47] both the characteristic slip length and the relaxation time are governed by the indentation depth and the dimensionless parameter vt/a just as the stationary properties.
5. Conclusion
In the present paper we discussed the question of whether it is possible to find such "robust governing parameters" which determine the friction most directly while remaining in the framework of a simple analytical structure. We argued that the property which most directly and robustly determines the contact configuration is the indentation depth, whereas the parameters of the contact configuration which are relevant for friction are the surface gradient and the
characteristic size of micro-asperities. Both parameters depend only on the indentation depth, though this dependence is relatively weak. Another parameter which is of importance is the characteristic relaxation time of the medium which only depends on temperature. The suggested structure of the coefficient of friction allows to apply the phenomeno-logical "generalized master curve procedure" by measuring the coefficient of friction at different velocities, temperatures and indentation depths and applying the shifting procedure. As the indentation depth is not a parameter which normally can be controlled in experiment, the suggested frictional law has to be completed by a contact mechanical derivation connecting the indentation depth with the normal force, shape of the bodies and their materials parameters.
Acknowledgement
The author acknowledges a valuable discussion with I. Argatov. This work was supported by Deutsche Forschungsgemeinschaft (DFG).
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Поступила в редакцию 13.06.2015 г.
Сведения об авторе
Valentin L. Popov, Prof. Dr. of Berlin University of Technology, v.popov@tu-berlin.de
